Calculus Examples

$\frac{d y}{d x} = \frac{e^{y}}{e^{x}}$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Multiply both sides by $\frac{1}{e^{y}}$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{1}{e^{y}} \cdot \frac{e^{y}}{e^{x}}$

Step 1.2

Simplify.

Tap for more steps...

Step 1.2.1

Combine.

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{1 e^{y}}{e^{y} e^{x}}$

Step 1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{1 e^{y}}{e^{y + x}}$

Step 1.2.3

Multiply $e^{y}$ by $1$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{e^{y}}{e^{y + x}}$

Step 1.2.4

Cancel the common factor of $e^{y}$ and $e^{y + x}$ .

Tap for more steps...

Step 1.2.4.1

Factor $e^{y + x}$ out of $e^{y}$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{e^{y + x} e^{y - (y + x)}}{e^{y + x}}$

Step 1.2.4.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1

Multiply by $1$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{e^{y + x} e^{y - (y + x)}}{e^{y + x} \cdot 1}$

Step 1.2.4.2.2

Cancel the common factor.

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{e^{y + x} e^{y - (y + x)}}{e^{y + x} \cdot 1}$

Step 1.2.4.2.3

Rewrite the expression.

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{e^{y - (y + x)}}{1}$

Step 1.2.4.2.4

Divide $e^{y - (y + x)}$ by $1$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = e^{y - (y + x)}$

Step 1.2.5

Apply the distributive property.

$\frac{1}{e^{y}} \frac{d y}{d x} = e^{y - y - x}$

Step 1.2.6

Subtract $y$ from $y$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = e^{0 - x}$

Step 1.2.7

Subtract $x$ from $0$ .

$\frac{1}{e^{y}} \frac{d y}{d x} = e^{- x}$

Step 1.3

Rewrite the equation.

$\frac{1}{e^{y}} d y = e^{- x} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{y}} d y = \int e^{- x} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Simplify the expression.

Tap for more steps...

Step 2.2.1.1

Negate the exponent of $e^{y}$ and move it out of the denominator.

$\int 1 {(e^{y})}^{- 1} d y = \int e^{- x} d x$

Step 2.2.1.2

Simplify.

Tap for more steps...

Step 2.2.1.2.1

Multiply the exponents in ${(e^{y})}^{- 1}$ .

Tap for more steps...

Step 2.2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 1 e^{y \cdot - 1} d y = \int e^{- x} d x$

Step 2.2.1.2.1.2

Move $- 1$ to the left of $y$ .

$\int 1 e^{- 1 \cdot y} d y = \int e^{- x} d x$

Step 2.2.1.2.1.3

Rewrite $- 1 y$ as $- y$ .

$\int 1 e^{- y} d y = \int e^{- x} d x$

Step 2.2.1.2.2

Multiply $e^{- y}$ by $1$ .

$\int e^{- y} d y = \int e^{- x} d x$

Step 2.2.2

Let $u_{1} = - y$ . Then $d u_{1} = - d y$ , so $- d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

Tap for more steps...

Step 2.2.2.1

Let $u_{1} = - y$ . Find $\frac{d u_{1}}{d y}$ .

Tap for more steps...

Step 2.2.2.1.1

Differentiate $- y$ .

$\frac{d}{d y} [- y]$

Step 2.2.2.1.2

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

$- \frac{d}{d y} [y]$

Step 2.2.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 2.2.2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.2.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int - e^{u_{1}} d u_{1} = \int e^{- x} d x$

Step 2.2.3

Since $- 1$ is constant with respect to $u_{1}$ , move $- 1$ out of the integral.

$- \int e^{u_{1}} d u_{1} = \int e^{- x} d x$

Step 2.2.4

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$- (e^{u_{1}} + C_{1}) = \int e^{- x} d x$

Step 2.2.5

Simplify.

$- e^{u_{1}} + C_{1} = \int e^{- x} d x$

Step 2.2.6

Replace all occurrences of $u_{1}$ with $- y$ .

$- e^{- y} + C_{1} = \int e^{- x} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Let $u_{2} = - x$ . Then $d u_{2} = - d x$ , so $- d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 2.3.1.1

Let $u_{2} = - x$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 2.3.1.1.1

Differentiate $- x$ .

$\frac{d}{d x} [- x]$

Step 2.3.1.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 2.3.1.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.3.1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- e^{- y} + C_{1} = \int - e^{u_{2}} d u_{2}$

Step 2.3.2

Since $- 1$ is constant with respect to $u_{2}$ , move $- 1$ out of the integral.

$- e^{- y} + C_{1} = - \int e^{u_{2}} d u_{2}$

Step 2.3.3

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$- e^{- y} + C_{1} = - (e^{u_{2}} + C_{2})$

Step 2.3.4

Simplify.

$- e^{- y} + C_{1} = - e^{u_{2}} + C_{2}$

Step 2.3.5

Replace all occurrences of $u_{2}$ with $- x$ .

$- e^{- y} + C_{1} = - e^{- x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$- e^{- y} = - e^{- x} + K$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Divide each term in $- e^{- y} = - e^{- x} + K$ by $- 1$ and simplify.

Tap for more steps...

Step 3.1.1

Divide each term in $- e^{- y} = - e^{- x} + K$ by $- 1$ .

$\frac{- e^{- y}}{- 1} = \frac{- e^{- x}}{- 1} + \frac{K}{- 1}$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- y}}{1} = \frac{- e^{- x}}{- 1} + \frac{K}{- 1}$

Step 3.1.2.2

Divide $e^{- y}$ by $1$ .

$e^{- y} = \frac{- e^{- x}}{- 1} + \frac{K}{- 1}$

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Dividing two negative values results in a positive value.

$e^{- y} = \frac{e^{- x}}{1} + \frac{K}{- 1}$

Step 3.1.3.1.2

Divide $e^{- x}$ by $1$ .

$e^{- y} = e^{- x} + \frac{K}{- 1}$

Step 3.1.3.1.3

Move the negative one from the denominator of $\frac{K}{- 1}$ .

$e^{- y} = e^{- x} - 1 \cdot K$

Step 3.1.3.1.4

Rewrite $- 1 \cdot K$ as $- K$ .

$e^{- y} = e^{- x} - K$

Step 3.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- y}) = \ln (e^{- x} - K)$

Step 3.3

Expand the left side.

Tap for more steps...

Step 3.3.1

Expand $\ln (e^{- y})$ by moving $- y$ outside the logarithm.

$- y \ln (e) = \ln (e^{- x} - K)$

Step 3.3.2

The natural logarithm of $e$ is $1$ .

$- y \cdot 1 = \ln (e^{- x} - K)$

Step 3.3.3

Multiply $- 1$ by $1$ .

$- y = \ln (e^{- x} - K)$

Step 3.4

Divide each term in $- y = \ln (e^{- x} - K)$ by $- 1$ and simplify.

Tap for more steps...

Step 3.4.1

Divide each term in $- y = \ln (e^{- x} - K)$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\ln (e^{- x} - K)}{- 1}$

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\ln (e^{- x} - K)}{- 1}$

Step 3.4.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (e^{- x} - K)}{- 1}$

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Move the negative one from the denominator of $\frac{\ln (e^{- x} - K)}{- 1}$ .

$y = - 1 \cdot \ln (e^{- x} - K)$

Step 3.4.3.2

Rewrite $- 1 \cdot \ln (e^{- x} - K)$ as $- \ln (e^{- x} - K)$ .

$y = - \ln (e^{- x} - K)$

Step 4

Simplify the constant of integration.

$y = - \ln (e^{- x} + K)$